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AP Calculus: Innovations

Redefining Calculus

Technology was not the only reform idea to find its way into AP Calculus classrooms in the early 1990s; many AP teachers were already adapting their courses to include cooperative learning, the constructive approach to teaching, the Rule of Three (or Four), and a heavier emphasis on concepts and on modeling. Experienced AP teachers also began to notice a change in the flavor of the exams, as the questions became less predictable and formulaic. These changes were subtle and gradual for the most part, mainly because the AP Development Committee had made a deliberate decision back in 1989 -- when laying out the six-year plan for the incorporation of graphing calculator technology into the AP Calculus course -- that they would not make substantial changes in the course description itself until the technology had been in place for three years. That meant that the target date for having a new AP Calculus curriculum in place was to be the 1997-98 school year. As everyone reading these words probably realizes, that new curriculum is now comfortably in place and will define the calculus experience of our best students for years to come.

Redefining AP Calculus, Part I
Redefining AP Calculus, Part II

Redefining AP Calculus, Part I
There have been three times that the AP Calculus Development Committee has had to start essentially from scratch in defining what would constitute an AP course in mathematics.

The first time was at the outset, when the Brinkman committee (CAAS: College Admission with Advanced Standing) decided to make it a yearlong course in single-variable calculus. Once that decision had been made, the rest was pretty easy: all one had to do was open up a mainstream textbook -- which textbook you opened hardly mattered -- and copy the table of contents. (It is doubtful that they actually did that, although the resulting course description as published for many years certainly looked like a table of contents, making it an easy target for reformers.)

The second genesis for AP Mathematics came in 1967 when the Development Committee, chaired by Al Tucker of Princeton, set out to invent Calculus AB and Calculus BC. (The first AB and BC exams were given in 1969.) The program had grown to the point that AP Mathematics seemed to be unnecessarily ambitious for some students seeking credit, placement, or both at some colleges. So a less ambitious option was envisioned that would be equivalent to at least a semester of calculus at any college or university, and perhaps equivalent to a year at others if you could get them to admit it.

Calculus BC turned out to be easy to define; it was essentially the AP course that had been in place for more than a decade. Calculus AB had to be built from the ground up (after, indeed, the ground had been identified), and that was not so easy to do. Fortunately, the CUPM (Committee on the Undergraduate Program in Mathematics) had already undertaken an extensive study of the same issues and had made their recommendations, enabling the Development Committee to build upon their framework.

An unexpectedly vexing problem proved to be the naming of the two courses. The CUPM had broken the material for the first two years of college mathematics into sections called 0, 1, 2, and 3. The Committee designed the less ambitious courses to cover roughly 0, 1, and 2, while the more ambitious course would cover roughly 1, 2, and 3. These numbers were too suggestive of AP grades to be useful for naming the courses, so eventually they retained the concepts of sequence and overlap by using the letters A, B, and C.

For the next 30 years the course descriptions hardly changed at all, even though they were under constant review by a committee with a constantly-changing membership. Little topics would come and go, or drift from the AB column to the BC column or vice versa, but that was about it. Typical was the saga of L'H˘pital's rule, which began as a BC topic. The Development Committee discovered through conversations among teachers at the AP Reading that AB teachers were teaching it anyway, as a little trick to handle those ugly limit problems. Concerned that their ugly limit problems were therefore not discriminating effectively, the Committee decided to level the playing field by moving L'H˘pital's rule to the AB column, where it remained for many years. The 1998 course description finally abandoned the ugly limit problems, sending L'H˘pital back to the BC curriculum, where the rule that bears his name is actually useful: for finding improper integrals and testing series for convergence.

Redefining AP Calculus, Part II
Three decades of this sort of tinkering had a gradual disheveling effect on the course descriptions, which by 1990 lacked even the logical organization of a reputable table of contents. As noted earlier, the AP Calculus Development Committee had wanted to avoid confusing the teachers by introducing curriculum changes at the same time that students were adjusting to the calculators. Nonetheless, in 1994 they did take the intermediate step of reorganizing the course descriptions in a more logical order, eliminating redundancies and clarifying topics that had been known to cause confusion. Since the courses still covered the same topics, this cosmetic improvement did not constitute a new definition of AP Mathematics.

It was clearly only a matter of time, however, before the Committee would have to redefine AP Calculus for the third time. Anita Solow participated in the Calculus for a New Century conference in 1987 and remained active in the reform movement from its inception. She later edited several important calculus reform publications for the Mathematical Association of America, but more significantly for this bit of history, she became involved in the AP Program as a reader and table leader in the 1980s, then joined the Development Committee in 1992. Her early years with that group were devoted to the smooth transition to graphing calculators, but questions of calculus course content were never far below the surface.

A forum of College Board and ETS dignitaries and 23 of the most prominent figures in the world of calculus reform were called together in 1994 for the purpose of advising the Committee on what directions they should take when reforming the AP curriculum. In retrospect, it was an extraordinary gathering. While all of the principles had been well aware of each other's contributions to the evolution of calculus reform, the realities of the marketplace had, up to that point, discouraged any such open forum for sharing their conclusions and synthesizing their ideas. They may not have agreed on every curricular matter under discussion, but when they did agree there was electricity in the room. The Committee took careful notes, thanked the participants very much, rolled up their sleeves, and within five months produced Draft One of the new AP Calculus Course Descriptions.

The course descriptions went through several more drafts over the next 12 months following scrutiny by many diverse constituencies, including the participants in the October conference, Readers at the 1995 AP Reading, and AP teachers at workshops across the country the following fall. A "preliminary" draft of the course description was widely distributed in 1996, but even that changed slightly by the time the official version was released in 1997. A new Teacher's Guide for AP Calculus was released at that same time and was mailed free of charge to every AP Calculus high school in the country; such was the College Board's commitment to the successful implementation of the new reforms.

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